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« Uncalibrated Vision based on Structured Light »

David Fofi 1. « Uncalibrated Vision based on Structured Light ». Joaquim Salvi 2. El Mustapha Mouaddib 3. 1 Le2i UMR CNRS 5158 Université de Bourgogne Le Creusot, France d.fofi@iutlecreusot.u-bourgogne.fr. 2 VICOROB - IIiA Universitat de Girona Girona, España qsalvi@eia.udg.es.

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« Uncalibrated Vision based on Structured Light »

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  1. David Fofi1 « Uncalibrated Vision based on Structured Light » Joaquim Salvi2 El Mustapha Mouaddib3 1Le2i UMR CNRS 5158 Université de BourgogneLe Creusot, Franced.fofi@iutlecreusot.u-bourgogne.fr 2VICOROB - IIiAUniversitat de GironaGirona, Españaqsalvi@eia.udg.es 3CREA EA 3299Université de Picardie Jules VerneAmiens, Francemouaddib@u-picardie.fr

  2. 0. Outline …………………………… • Introduction • Tools for uncalibrated vision • Uncalibrated reconstruction • Experimental results • Conclusion

  3. I. Introduction …………………………… • Structured light vision • Calibration vs uncalibration

  4. IMAGE PLANE PATTERN FRAME « Structured light vision » ……………………...…………………………… ……………….. J. Salvi, J. Batlle, E. Mouaddib, "A robust-coded pattern projection for dynamic measurement of moving scenes", Pattern Recognition Letters, 19, pp. 1055-1065, 1998. J. Batlle, E. Mouaddib, J. Salvi, "Recent progress in coded structured light to solve the correspondence problem. A survey", Pattern Recognition, 31(7), pp. 963-982, 1998.

  5. « Calibration vs uncalibration » ……………………...…………………………… • DRAWBACKS OF HARD-CALIBRATION: • Off-line process (calibration pattern, etc.) • Has to be repeated each time one of the parameters is modified Working with a camera with automatic focus and aperture is NOT possible. Visual adaptation to the environment is not allowed! A slide or LCD projector needs to be focused. RECONSTRUCTION FROM UNCALIBRATED SENSOR...

  6. II. Tools for uncalibrated vision …………………………… • Test of spatial colinearity • Test of coplanarity • Stability of the cross-ratio • Validity of the affine model

  7. S R Q P « Test of spatial colinearity » ……………………...…………………………… Cross-ratio within the pattern and cross-ratio within the image are equals if the points are colinear.

  8. « Test of coplanarity » ……………………...…………………………… o p o' p' s' s r' q' r q {o;p,q,r,s}={o';p',q',r'} Cross-ratio within the pattern and cross-ratio within the image are equals if the point are colinear.

  9. « Stability of the cross-ratio » ……………………...…………………………… Error on cross-ratios with a noise from 0 to 0.5d (d is the distance between two successive points) Nota: to compare cross-ratios a projective distance is necessary. Method of the random cross-ratios. ……………….. K. Aström, L. Morin, "Random cross-ratios", Research Report n°rt 88 imag-14, LIFIA, 1992.

  10. n m n' m' « Validity of the affine model » ……………………...…………………………… affine projection Valid if 0

  11. III. Uncalibrated reconstruction …………………………… • Projective reconstruction • Structured light limitations • Euclidean constraints through structured lighting

  12. « Projective reconstruction » ……………………...…………………………… Recover the scene structure from n images and m points and...  Intrinsic parameters Extrinsic parameters Scene geometry Points matching    PROJECTIVE RECONSTRUCTION

  13. MOVEMENT OF THE PROJECTOR CAMERA + PROJECTOR THE PATTERN SLIDES ALONG THE OBJECTS HETEROGENEITY OF THE SENSOR + MOVEMENT OF THE 3-D POINTS INTRINSIC PARAMETERS CANNOT BE ASSUMED CONSTANT RECONSTRUCTION FROM TWO VIEWS (i.e. one view and one pattern projection) = « Structured light limitations » ……………………...…………………………… PARAMETERS ESTIMATION APPROACH, CANONICAL REPRESENTATION ……………….. R. Mohr, B. Boufama, P. Brand, “Accurate projective reconstruction”, Proc. of the 2nd ESPRIT-ARPA-NSF Workshop on Invariance, Azores, pp. 257-276, 1993. Q.-T. Luong, T. Viéville, "Canonical representations for the geometries of multiple projective views", Proc. of the 3rd Euro. Conf. on Computer Vision, Stockholm (Sweden), 1994

  14. « The parameters estimation approach » ……………………...…………………………… n images composed by m points... pij : image point Aj : projection matrix Pj : object point (Uij, Vij) : pixel co-ordinates

  15. « The parameters estimation approach » ……………………...…………………………… A unique solution cannot be performed because... W is a 4x4 invertible matrix… a collineation of the 3-D space 4x4 - 1 (scale factor) = 15 degrees of freedom, thus... 5 co-ordinates object points assigned to AN ARBITRARY PROJECTIVE BASIS. A RECONSTRUCTION WITH RESPECT TO A PROJECTIVE FRAME (distances, angles, parallelism are not preserved)

  16. « From projective to Euclidean » ……………………...…………………………… Euclidean transformations form a sub-group of projective transformations... A collineation W upgrades projective reconstruction to Euclidean one. TRANSLATING EUCLIDEAN KNOWLEDGE OF THE SCENE INTO MATHEMATICAL CONSTRAINTS ON THE ENTRIES OF W. Matching projective points with their corresponding Euclidean points ? YES, BUT ... Euclidean co-ordinates of points are barely available… … if they are: pattern cross-points have to be projected exactly onto these object points. ……………….. B. Boufama, R. Mohr, F. Veillon, "Euclidean constraints for uncalibrated reconstruction", Proc. of the 4th Int. Conf. on Computer Vision, Berlin (Germany), pp. 466-470, 1993.

  17. A  B  D  C  ONTO A PLANAR SURFACE IMAGE CAPTURE PROJECTED SQUARE « Parallelogram constraints » ……………………...…………………………… The sensor behaviour is assumed to be affine...

  18. Vert. plane Pattern Horiz. plane « Alignment constraints » ……………………...…………………………… • Arbitrary distance between two planes... • Points belonging to horizontal or vertical plane... • Cross-point as origin…

  19. C' Light stripes Light planes B' Planar surfaces A' C A B Projected lines « Orthogonality constraints » ……………………...…………………………… A'B' ·A'C' = (xA' - xB')(xA' - xC')+ (yA' - yB')(yA' - yC')+ (zA' - zB')(zA' -zC') = 0 otherwise… reduced orthogonality constraint: (xA' - xB')(xA' - xC')+ (yA' - yB')(yA' - yC') = 0

  20. « Example » ……………………...…………………………… An alignment constraint : xA' = xB' (relation between unknown Euclidean points) We have: [xA' ; yA' ; zA' ; tA']T = W· [xA ; yA ; zA ; tA]T[xB' ; yB' ; zB' ; tB']T = W· [xB ; yB ; zB ; tB]T Then: W1i·xA = W1i·xB(relation between known projective points) … same way for the other constraints… The set of equations is solved by a non-linear optimisation method as Levenberg-Marquardt. 15 independent constraints are necessary (W is a 44 matrix defined up to a scale factor)

  21. IV. Experimental results …………………………… • Colinearity • Coplanarity • Euclidean reconstruction

  22. « Colinearity » ……………………...…………………………… Theoretical (pattern) cross-ratio = 1.3333 Measured (image) cross-ratio = 1.3287 Projective error =6.910-4 Decision = the points are colinear Theoretical cross-ratio = 1.3333 Measured cross-ratio = 1.3782 Projective error =6.210-3 Decision = the points are not colinear

  23. « Coplanarity » ……………………...…………………………… Theoretical cross-ratio = 2 Measured cross-ratio = 1.96 Projective error =2.210-3 Decision = the points are coplanar Theoretical cross-ratio = 2 Measured cross-ratio = 2.186 Projective error =5.910-3 Decision = the points are not coplanar

  24. « Euclidean reconstruction: synthetic data » ……………………...…………………………… re-projection of 3D points onto the image planes (circles: synthetic points, crosses: re-projections)

  25. -20 -40 -60 y -80 -100 -120 -140 -180 -160 -140 -120 -100 -80 -60 x « Euclidean reconstructions » ……………………...……………………………

  26. V. Conclusion ……………………………

  27. Projective reconstruction from a single pattern projection and a single image capture. • Pattern projection used to retrieve geometrical knowledge of the scene: uncalibrated Euclidean reconstruction. • Structured lighting ensures there is known scene structure. • Structured light provides numerous contraints. • Tests of colinearity and coplanarity can be used to retrieve projective basis (5 points, no 4 of them being coplanar, no 3 of them being colinear).

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